Set both equations equal to each other. 2=4cos(θ) θ=π/3 or 5π/3. θ=π/3 is the first place both graphs will meet. Integrating the difference of the areas of the functions from θ=0 to θ=π/3 and then doubling this area will give you the area you are looking for.
You want to integrate from θ=0 to θ=π/3 2 * 1/2 Integral[ (4*cosθ)^2 -2^2 dθ]